Take your pick of London housing shortage news. Jenny Jones is the latest reliable supplier. In a new report the Green party member of the London assembly, who also chairs its planning and housing committee, names "three trends in the past decade that have made housing increasingly difficult for Londoners to afford". One, a huge drop in the number of homes for social rent. Two, a huge rise in the cost of homes to buy. Three, the supply of new homes in London failing to match Londoners' needs.

The rest of the story falls gloomily into place. Boroughs' housing waiting lists have lengthened. Over-crowding is rife. More and more people on even fairly fat incomes have no hope of buying. Private renting too is beyond the range of many. The capital's population is unlikely to decrease. Housing supply of every kind seems to lag ever further behind demand and need.

This picture forms the big backdrop to recent grappling between Boris Johnson and his Labour foes. It kicked off when the mayor took advantage of a photo opportunity on a new doorstep in Charlton, and claimed to have "delivered" more than "20,000 extra affordable homes", calling this the greatest number in "any mayoral term". The Labour party in the London assembly accused him of taking credit for homes "started long before he was in City Hall".

Were they right? Depends what you mean by "started". Depends what you mean by "delivered". If by "started" you mean when planning permission was secured or projects got under way, then Boris's figure is a fraud and many of those 20,000 have only been completed thanks to government "kick start" money supplied to compensate for the credit crunch.

City Hall argues, though, that the mayor's policies and personal touch have meant that this and other funding provided through the Homes and Communities Agency (whose London board Boris chairs), has been to spend to innovative best effect, for example by bringing homes originally built for private sale into the "intermediate" price range through new "part buy, part rent" arrangements or by enabling closer engagement with housing associations. It protests that although he's had to let go his election target of 50,000 new "affordable" homes by April 2011, he's made the best of bad economic times.

It's too soon to really judge the whole of Boris's approach. Its most controversial part has been the negotiation of individual delivery targets with each borough. The theory is that "top down" direction – Ken Livingstone's "50% rule" hindered rather than helped provision and some boroughs complain that the richer ones have got off far too lightly. (Although, as one of the policy's biggest critics, Newham's Labour mayor Sir Robin Wales, observed to me, if builders aren't building the targets mean nothing anyway.)

Housing minister John Healey accused the mayor of lack of ambition – surely a first for this most voracious of politicians – and under-spending. And suspicions linger that his approach, like that of Tory flagship authorities and any future national Tory policy, will nurture nimbyism and hurt the least well-off.

Other measures have been praised from a wide range of interested parties. London assembly's Labour group leader Len Duvall has many criticisms yet has backed Boris's support, if only in the public sector, for a return to the Parker Morris space standards that Margaret Thatcher did away with (in the name of freedom, naturally). My own borough, the Labour stronghold of Hackney, has welcomed the chance to pilot further devolved powers. The Empty Homes Agency has applauded his allocation of £60m. The principles of Boris's strategy largely anticipate those that a Cameron government seems set to adopt, so far as we can tell: more localism, more incentives, a bit more help on to the housing ladder for some. But the sense remains that the powers of any London mayor to radically alter that big, metropolitan backdrop picture – gross overcrowding, long waiting lists and a private sale market gone mad – is limited. Perhaps more could be done to deal with the empty homes problem. Perhaps bold ideas such as community land trusts should be pursed with greater vigour. The quality of big city life depends so much on the homes its citizens inhabit. We need big, new ideas, and soon.